Italian, 2-rainbow and Roman domination numbers in middle graphs

Department of Mathematics Education, Daegu Catholic University, Gyeongsan-si 38430, Republic of Korea

^* Corresponding author: kkim@cu.ac.kr

Received: 20 April 2023
Accepted: 21 March 2024

Abstract

Given a graph G, we consider the Italian domination number γ_I(G), the 2-rainbow domination number γ_r2(G) and the Roman domination number γ_R(G). It is known that γ_I(G) ≤ γ_r2(G) ≤ γ_R(G) holds for any graph G. In this paper, we prove that γ_I(M(G)) = γ_r2(M(G)) = γ_R(M(G)) = n for the middle graph M(G) of a graph G of order n, which gives an answer for an open problem posed by Chellali et al. [Discrete Appl. Math. 204 (2016) 22–28]. Moreover, we give a complete characterization of Roman domination stable middle graphs, 2-rainbow domination stable middle graphs and Italian domination stable middle graphs.